Lorentz and Diffeomorphism Violations in Linearized Gravity

Lorentz and diffeomorphism violations are studied in linearized gravity using effective field theory. A classification of all gauge-invariant and gauge-violating terms is given. The exact covariant dispersion relation for gravitational modes involving operators of arbitrary mass dimension is constructed, and various special limits are discussed.

The foundational symmetries of General Relativity (GR) include diffeomorphisms and local Lorentz transformations. The former act on the spacetime manifold, while the latter act in the tangent space. These two types of transformations are partially linked through the vierbein, which provides a tool for moving objects between the manifold and the tangent space. The proposal that Lorentz invariance might be broken in an underlying theory of gravity and quantum physics such as strings [1,2] naturally raises various questions about the relationship between diffeomorphism violation and Lorentz violation and about the associated phenomenological signals. These questions can be studied independently of specific models using gravitational effective field theory [3]. Here, following a brief summary of the current status and results, we develop a model-independent framework for studying these issues in linearized gravity. This limit provides a comparatively simple arena for exploration, and it is crucial for experimental analyses of gravitational waves and of gravitation in the Newton and post-Newton limits.
A generic treatment of Lorentz violation in Minkowski spacetime in the absence of gravity is comparatively straightforward using effective field theory [4]. In this context, the role of diffeomorphisms and local Lorentz transformations is played by translations and Lorentz transformations that act globally and combine to form the Poincaré group. The two symmetries can be broken independently, and a physical breaking of either one can be represented in terms of nonzero background fields in an effective field theory. The breaking of either can be spontaneous or explicit. Spontaneous breaking occurs when the background is dynamical, which means that it must satisfy the equations of motion and that it comes with fluctuations in the form of Nambu-Goldstone modes [5] and possibly also massive modes. In most applications of spontaneous breaking, the background satisfies the equations of motion in vacuum and can therefore be viewed as the vacuum expectation value. In contrast, explicit breaking is a consequence of a prescribed background, which is typically off shell and has no associated fluctuations. Much of the phenomenological literature investigating Lorentz violation in Minkowski spacetime assumes for simplicity that global spacetime translations are preserved in an approximately local inertial frame, canonically taken to be the Sun-centered frame [6]. This guarantees conservation of energy and momentum, so phenomenological signals are restricted to violations of the conservation laws for generalized angular momenta.
A large body of experimental studies constrains this type of Lorentz violation [7].
In the presence of gravity, the situation becomes more involved. One complication arises because diffeomorphisms and local Lorentz transformations act on objects in different spaces that can be linked via the vierbein, which can relate the corresponding violations. In the case of spontaneous breaking, for example, the vacuum expectation values are on shell and a nonzero background on the spacetime manifold implies one in the tangent space and vice versa. As a result, diffeomorphism violation occurs if and only if local Lorentz violation does [8]. More intuitively, local Lorentz violation can be understood as a background direction dependence in a local freely falling frame [3]. Transporting this to the spacetime manifold via the vierbein then guarantees the existence of a direction dependence on the spacetime manifold and hence diffeomorphism violation.
Another complication for gravity concerns conservation laws and arises from the difference between spontaneous and explicit breaking. In general, a theory invariant under local transformations comes with covariantly conserved currents [9]. In spontaneous breaking, the full theory remains invariant under the transformations and the symmetry is only hidden [10]. The currents remain conserved even though the background is unchanged by the transformations because the background fluctuations transform in a nonstandard way to compensate. This contrasts with explicit breaking, when the current conservation laws fail to hold.
In GR, local Lorentz invariance implies symmetry of the energy-momentum tensor while diffeomorphism invariance implies its covariant conservation [11]. In theories with spontaneous diffeomorphism and local Lorentz violation, these current-conservation laws are unaffected: an energy-momentum tensor for the full theory remains covariantly conserved and it is always possible to make it symmetric [3]. However, if explicit breaking occurs, then there is no guarantee that the energy-momentum tensor is explicitly conserved or symmetric, and as a result a theory with explicit breaking can be inconsistent or require reformulation within Finsler geometry [3,12]. For sufficiently involved models, this situation can be rescued by the additional modes that appear in theories with explicit diffeomorphism and local Lorentz violation [13]. These additional modes arise because in explicit breaking it becomes impossible to remove all four diffeomorphism degrees of freedom and six local Lorentz degrees of freedom from the vierbein. In some models, these additional modes can be constrained to restore the covariant conservation and symmetry of the energy-momentum tensor. The additional modes are the counterparts in explicit breaking of the Nambu-Goldstone modes appearing in spontaneous breaking. Indeed, they can be understood as Nambu-Goldstone excitations of Stueckelberg fields [14,15].
The above results have several implications for the phenomenology of diffeomorphism and local Lorentz violations in gravity. If the breaking is explicit, the challenge lies in establishing the consistency of theory and, if achieved, then in determining the effects of the additional modes on observational signals. In contrast, if the breaking is spontaneous, the Nambu-Goldstone and massive fluctuations can play the role of new forces affecting the phenomenology and so must be taken into account in analyzing experimental signals.
An alternative model-independent approach to studing both spontaneous and explicit diffeomorphism and local Lorentz violation uses linearized effective field theory for gravity, formulated to incorporate gauge and Lorentz violation [46]. In this context, gauge transformations are linearized diffeomorphisms of the metric fluctuation. This technique yields an explicit construction and classification of the general quadratic Lagrange density in effective field theory with gauge invariance at linearized level. It also permits construction of the general covariant dispersion relation and investigation of the properties of the corresponding gravitational modes. These results have been applied to obtain model-independent constraints on linearized coefficients for Lorentz violation using gravitational waves [46,51] and tests of gravity at short range [52,53]. In the present work, we extend this approach to explicit gauge breaking. We construct and classify all possible terms for the quadratic Lagrange density in gravitational effective field theory with explicit gauge violation, and we derive the corresponding covariant dispersion relation required for experimental applications.
To perform the linearization, we expand the dynamical metric g µν in a flat-spacetime background with Minkowski metric, g µν = η µν + h µν . A generic term of mass dimension d ≥ 2 in the Lagrange density for the linearized gravitational effective field theory can then be written as where where the upper sign holds for odd d and the lower one for even d.

The action is invariant under the usual gauge transformations
Assuming this condition, the operators K (d)µνρσ can be constructed explicitly, using standard methods in group theory [57].
They are found to span three representation classes [46]. For the present work, we have ex-  Table I. To obtain the term in the Lagrange density (1) associated to a given class, it suffices to replace K (d)µνρσ with the operator listed. The second column displays the index symmetries of each class using Young tableaux. The Table also lists some properties of each class. The third column indicates whether the operator is fully gauge invariant, and the fourth column displays the handedness under CPT of the associated term in the Lagrange density. Each class can occur only for even or for odd d and for d above a minimal value, as shown in the next column. The final column lists the total number of independent components appearing in the coefficient The quadratic approximation L 0 to the Lagrange density for the Einstein-Hilbert action can conveniently be written in the form This gauge-and Lorentz-invariant term is constructed from a piece of the coefficient  Table I can then be expressed as where the sum is over all the representation classes K (d)µνρσ shown in Table I  However, in the presence of partial or full gauge invariance, finding a dispersion relation for the physical modes is more complicated because M κλ µν contains a null space. Nonetheless, a covariant dispersion relation can be found using methods from exterior algebra, as we show next. The technique presented here is a generalization of the method developed by us for the study of the photon sector of the SME [58] and independently by Itin for studies of premetric electrodynamics [59].
The key idea is to treat h µν as an element of a 10-dimensional complex vector space and M κλ µν as a linear map on the space. To keep the discussion general, we work with an N-dimensional complex vector space V and the exterior algebra ∧V over V. In terms of an arbitrary set of basis vectors {v a }, a = 1, 2, . . . , N, we write an n-vector ω as and take its Hodge dual * ω to have components given by Given a linear map M : V → V taking an arbitrary vector x ∈ V to a vector y = M · x, we can construct a natural linear map ∧ n M between n-vectors. For n arbitrary vectors {x 1 , x 2 , . . . , x n }, we define In components this gives In particular, while the map M is rank r, the map ∧ r M is rank 1.
The dual map * ∧ r M can be constructed using the Hodge dual (6). However, both ∧ r M and * ∧ r M incorporate a null space, which complicates the derivation of the dispersion relation. To account explicitly for the null space of ∧ n M, we can work instead with a modified dual ⋆∧ n M . Introducing ζ = * (z 1 ∧ · · · ∧ z s ), some consideration reveals that we In the special case where n = r, we see that ⋆∧ r M is a scalar obeying ∧ r M = ⋆∧ r M ζ * ⊗ ζ, which implies The inverse relation for the modified dual can be obtained from Eq. (9). After some manipulation, we find For the case S = ∅ so that V = R, the modified dual ⋆∧ n M reduces to the usual dual * ∧ n M.
Taking instead r = n yields the modified scalar dual Note that since the modified scalar dual ⋆∧ r M contains inverse powers of ζ * · ζ, one might naively expect a singularity when ζ * · ζ = 0. However, continuity and the relation ∧ r M = ⋆∧ r M ζ * ⊗ ζ guarantee that ⋆∧ r M remains finite for nonzero ζ. Similarly, although ⋆∧ n M may contain divergences, they cannot contribute to ∧ n M in Eq. We seek nontrivial solutions to M · x = 0, which exist if we can find p µ that reduce the rank of M by at least one. The exact covariant dispersion relation arising from M · x = 0 can therefore be expressed as After expanding ⋆ ′ ∧ r ′ M in terms of M 0 and δM, some calculation reveals that this equation can be written as The sum in this expression is understood to be limited to nonnegative wedge powers and calculation then yields an alternative form for the dispersion relation, which holds for S ⊇ S ′ and is convenient when the form of the broken gauge vectors As an application of the above results, consider linearized GR. The linearized Einstein field equations take the form M 0µν ρσ h ρσ = 0. The operator M 0µν ρσ can be expressed as where the projections P µν and π µν ρσ given by are a subset of the standard spin-2 projection operators [60,61]. Note that the trace of P is 3, while that of π is 6. The wedge products are found to be ∧ n M 0 = p 2n 2 n ∧ n π − n(∧ (n−1) π) ∧ (P ⊗ P ) , (19) and the scalar dual is The first term in the brackets is just ζ * · ζ. The second term gives −ζ * · ζP · P = −ζ * · ζtrP = −3ζ * · ζ. The four gauge vectors can be written as (z κ ) µν = (Z κ ) (µ p ν) , κ = 1, . . . , 4, where (Z κ ) µ are four independent vectors. This implies ζ * · ζ = 6! 2 5 |Z| 2 p 8 , where Z = det(Z κ α ). Putting together the pieces yields The dispersion relation for linearized GR is thus found to be p 4 = 0, matching the standard result.
Next, consider the case where linearized GR is corrected by generic gauge-invariant terms.
This case has been explicitly treated in Ref. [46]. We write M = M 0 +δM, where M 0 is given by Eq. (17) and δM has at least the gauge invariances of M 0 but is otherwise arbitrary. We can simplify this case by noting that the factor π − P ⊗ P of projection operators appearing in M 0 obeys (π − P ⊗ P ) · (π − 1 2 P ⊗ P ) = π. The combination ̟ = π − 1 2 P ⊗ P can therefore be viewed as the gauge-invariant inverse of π − P ⊗ P , which suggests defining where I − 1 2 η ⊗ η is the trace-reversal operator. A short derivation then reveals that so the dispersion relation for M is the same as that forM . Direct calculation gives With the reasonable assumption that higher-order terms remain finite as p 2 → 0, this implies the leading-order covariant dispersion relation where δM n = tr(δM n ). The solution for the two perturbative modes is Upon explicit evaluation of the traces for the general gauge-invariant terms listed in Table   I, this equation reduces correctly to the results (5) and (6) for the dispersion relation given in Ref. [46].
With these previously known examples reproduced, we turn to the case where linearized GR is instead corrected by arbitrary gauge-violating terms. The covariant dispersion relation takes the form (13) with S ⊇ S ′ , so to evaluate it explicitly we must determine the modified duals that appear in Eq. (15). They are After some calculation, we find the covariant dispersion relation to be The number of terms in the sum is restricted by the rank δr of δM, which may be less than the total rank r ′ . The limits on the sum are thus 0 ≤ n ≤ δr − s.
The result (28)  Since that term is proportional to p 4 , we find the striking result that all models of this type must leave unaffected the conventional dispersion relation p 2 = 0.
When the rank δr of δM is one, this result reduces to the monomial p 4 z * δM z = 0 and so gives the usual dispersion relation p 2 = 0, in agreement with the conclusion above. However, when δr > 1, the dispersion relation becomes a polynomial, and the solutions can describe modifications to the behavior of the gravitational modes.
Since the covariant dispersion relation (28)  Another interesting special case is the set of models with coefficients having only purely temporal components, which is a subset of the isotropic limit. All gauge-invariant models of this type are discussed in Ref. [46]. For  vector exists, which can be taken as z µν = η 0(µ p ν) , so s = 1 as well. The dispersion relation becomes p 4 d k (d,1)00... E d = 0 and reduces to p 2 = 0, in agreement with the general result for δr = s discussed above. These coefficients therefore have no effect on the behavior of the usual gravitational modes.
More involved cases exist that also leave unaffected the usual gravitational modes. One example with δr = s = 2 involves the CPT-odd operator q (d,3)µρ•νσ• d−3 for d = 3. Taking the dual of this and restricting to the purely temporal coefficient q (3,3)000 produces a ranktwo matrix δM with nonvanishing components δM 0j0k = −i q (3,3)000 ǫ jkl p l /8 that is symmetric under interchange of the first or second pair of indices and is antisymmetric under interchange of the pairs. Two unbroken gauge vectors exist, which can be taken as (z ′ 0 ) µν = η 0(µ p ν) and (z ′ 1 ) µν = p µ p ν . Calculation shows the dispersion relation is p 4 ( q (3,3)000 ) 2 E 4 = 0 and so again yields p 2 = 0, as expected. Other components of q (3,3)µρλνσ can, however, modify gravitational propagation [62].
To summarize, we have provided in this work a framework for studying diffeomorphism